Textbook Thermal Physics (Danil V. Schroder), pages 150-151. I have a problem with the following extract:
“Helmholtz free energy ($F = U – TS$) is the total energy needed to create the system, minus the heat you can get for free from an environment at temperature $T$. This heat is given by $TΔS = TS$, where $S$ is the system’s (final) entropy; …”.
Let’s think that the system is an ideal gas, which will be created in a closed cylinder that is in thermal contact with the environment at temperature $T$.
So, $TΔS = TS$. Then the initial entropy of the system is zero, because the system doesn’t exist yet, right?
Then we start creating the system (its particles) at nearly zero temperature (still zero entropy ?). Appearing in the environment at temperature $T$, the system will absorb heat from the environment. This will increase thermal energy of the system so that its temperature rises from 0 to T. If it’s correct, we can write $TΔS = U_{\rm th}$. Consequently, $F$ – is work needed to make non-thermal internal energy of the system (chemical and nuclear energy). Then, we can write $$F = U - U_{\rm th}$$
(Interesting thing, in textbook Principles & Practice of Physics by Eric Mazur these energies and something else are grouped to so called source energy. So, if all the above is correct, I guess one can write $F = U_{\rm source}$.)
- Why is this heat equal to $TΔS$? The process is isothermal for the environment but not for the system. So, we can use the simplest expression for entropy change for the environment only: $$ΔS_{\rm env} = -\frac{Q}{T}$$ but we cannot use it for the system. Here $ΔS_{\rm env}$ is not equal to $-ΔS_{\rm sys}$ because the process doesn’t seem to be reversible.
Where I am wrong? Or how exactly the system should be created so that $F$ is just work to create this system and absorbed heat is equal to $TS_{\rm sys}$?